A note on the shape of the generalizedc-numerical range

Publisher Website
Google Scholar
Abstract
Let C=(cij )∈Mk(C), and let be a complex Hilbert space of dimension greater than or equal to k. Let be a bounded linear operator. The C-numerical range of T, denoted by Wc (T), is defined to be the set of points obtained by letting (e 1,…,ek ) vary over the set of all orthonormal k-tuples in . Many results concerning the shape of the C-numerical range have been obtained when C is normal, but few results are known about the shape of the C-numerical range in general. In this note we consider the shape of the C-numerical range. In particular, along with noting that the C-numerical range of an operator T is path-connected, we show that, in the case when is a complex infinite-dimensional Hilbert space, the closure of the C-numerical range of an operator T is star-shaped with respect to the set (trC)We (T), where We (T) denotes the essential numerical range of T. 1 1Thanks are due to Dr. H. P. Rogosinski for his supervision in the preparation of this paper. View all notes

This publication has 5 references indexed in Scilit: